NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7  Correlation
NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7  Correlation Page/Excercise 104
The correct answer is option (iii). If there is no relationship between the two variables, then r = 0. It indicates the nonexistence of correlation between height in feet and weight in kilograms.
The correct answer is option (ii). The range of simple correlation coefficient is minus one to plus one (±1). If r = +1, then there is perfect positive correlation. While r = 1, there is negative correlation between the two variables.
The correct answer is option (i). If r_{xy} is positive, then the relation between X and Y is of the type positive correlation, i.e. when Y variable increases, the variable X also increases. In this case, the two variables move in the same direction.
NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7  Correlation Page/Excercise 105
The correct answer is option (ii). If r_{xy} = 0, the variables X and Y are uncorrelated. There is no linear relation between them. However, they may be related to each other nonlinearly.
The correct answer is option (iii). A scatter diagram provides a visual idea about the nature of association between the two variables. It is the simplest method of studying the relationship between two variables without calculating any numerical value. Karl Pearson's coefficient of correlation is to be applied only when the deviation of items is taken from actual means and not from assumed means. Similarly, Spearman's rank correlation can be used for calculating rank correlation only for quantitative data.
The correct answer is option (i). If precisely measured data are available, the simple correlation coefficient is more accurate than the rank correlation coefficient. All the properties of the simple correlation are applicable to Spearman's method. However, it is not as accurate as Karl Pearson's method of correlation because all the information concerning the data is not used.
Correlation coefficient r and covariance measure the degree of linear relationship between variables X and Y. However, the correlation coefficient is generally preferred to covariance because
 r has no unit and is a pure number.
 It is independent of origin and scale.
 Its value lies between 0 and 1.
No, the value of correlation coefficient (r) cannot lie outside the 1 and 1 range depending on the type of data. However, if the value of r lies outside this range, then it implies error in the estimation of correlation coefficient.
No, correlation does not measure causation. This is because it measures the degree and intensity of relationship between two variables. Even a high degree of correlation does not necessarily indicate causation between the variables. So, it is necessary to ensure that the variables for correlation analysis are properly selected to make the analysis more useful.
Rank correlation is more precise than simple correlation coefficient in the following ways:
 Measurement of the variables: In some areas, a measuring rod or weighing machine is not used to measure the height and weight of people. So, quality information of people is used to assign rank in terms of height and weight.
 Measurement of qualitative data: Attributes such as intelligence, taste and discipline are difficult to quantify. Hence, ranking may be a better alternative to quantification of qualities.
 Presence of extreme values: Correlation coefficient between two variables with extreme values may be different without extreme values. Hence, ranking may be a better alternative to simple correlation.
No, zero correlation does not imply independence of two variables. If there is zero correlation, it means the two variables (say X and Y) are not correlated and there is no linear relation between them. However, some other type of relation exists between them.
No, simple correlation coefficient cannot measure any type of relationship because it only specifies the magnitude and direction of correlation, i.e. whether correlation is positive or negative. It only indicates that there is nonlinear relationship between two variables but does not measure it such as quadratic, trigonometric and cubic.
Please note that you would need to perform this activity and interpret the result accordingly.
Calculation of coefficient of two variables:
Height of Classmate X 
Height of Benchmate Y 
58 
62 
69 
66 
65 
58 
68 
67 
71 
69 
59 
70 
69 
71 
XSeries 
YSeries 
XY 

X 
X^{2} 
Y 
Y^{2} 

58 
3364 
62 
3844 
3596 
69 
4761 
66 
4356 
4554 
65 
4225 
58 
3364 
3770 
68 
4624 
67 
4489 
4556 
71 
5041 
69 
4761 
4899 
59 
3481 
70 
4900 
4130 
69 
4761 
61 
3721 
4209 
∑X = 459 
∑ X^{2 }= 30257 
∑Y = 453 
∑Y^{2 }= 29435 
∑XY = 29714 
The coefficient of correlation is 0.073. This implies low degree of positive correlation.
List of variables where accurate measurement is difficult:
 Qualitative variables such as
 Beauty and income earned through modelling
 Honesty and job opportunities in the market
 Subjective variables such as
 Poverty and growth of population
 Development and infrastructural facilities
 Correlation (r) = 1: If there is perfect positive relationship between two variables, then the value of correlation will be +1.
 Correlation (r) = 1: If there is perfect negative relationship between two variables, then the value of correlation will be 1.
 Correlation (r) = 0: If there is no relationship between the two variables, then the value of correlation will be zero. However, it does not imply that these two variables are independent. It only indicates nonexistence of linear relation between the two variables.
Rank correlation coefficient differs from Pearsonian correlation coefficient in the following ways:
 Karl Pearson's method of correlation measures correlation for quantitative data such as income and savings of the household, while Spearman's method of rank correlation measures correlation for qualitative data such as intelligence and beauty of a person.
 Karl Pearson's method of correlation computes deviations from actual or assumed mean, but Spearman's method of rank correlation measures the differences in rank.
 Karl Pearson's method of correlation has high significance to extreme values as it is based on actual values, while Spearman's method has low significance to extreme values as it provides them rank.
XSeries 
YSeries 
XY 

X 
X^{2} 
Y 
Y^{2} 

65 
4225 
67 
4489 
4355 
66 
4356 
56 
3136 
3696 
57 
3249 
65 
4225 
3705 
67 
4489 
68 
4624 
4556 
68 
4624 
72 
5184 
4896 
69 
4761 
72 
5184 
4968 
70 
4900 
69 
4761 
4830 
72 
5184 
71 
5041 
5112 
∑X = 534 
∑X^{2 }= 35788 
∑Y = 540 
∑Y^{2 }= 36644 
∑XY = 36118 
The correlation coefficient is 0.44.
XSeries 
YSeries 
XY 

X 
X^{2} 
Y 
Y^{2} 

3 
9 
9 
81 
27 
2 
4 
4 
16 
8 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
2 
4 
4 
16 
8 
3 
9 
9 
81 
27 
∑X = 0 
∑X^{2 }= 28 
∑Y = 28 
∑Y^{2 }= 196 
∑XY = 0 
Coefficient of correlation (r) is 0. This implies that there is no linear relation between X and Y.
NCERT Solution for Class 11 Commerce Statistics for Economics Chapter 7  Correlation Page/Excercise 106
XSeries 
YSeries 
XY 

X 
X^{2} 
Y 
Y^{2} 

1 
1 
2 
4 
2 
3 
9 
6 
36 
18 
4 
16 
8 
64 
32 
5 
25 
10 
100 
50 
7 
49 
14 
196 
98 
8 
64 
16 
256 
128 
∑X = 28 
∑X^{2 }= 164 
∑Y = 56 
∑Y^{2 }= 656 
∑XY = 328 
The correlation coefficient between X and Y is 1. Therefore, there is perfect positive relationship between two variables X and Y.